Two-fold branched covers

Many three-dimensional manifolds are two-fold branched covers of the three-dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and includes many examples. It shows that there are three-dimensional homology spheres that do not two-fold branched cover any manifold, ones that only two-fold branched cover the three-dimensional sphere, ones that just twofold branched cover a non-trivial manifold, and ones that two-fold branched cover both the sphere and non-trivial manifolds. When a manifold is surgery on a knot, the possible quotients via involutions generically correspond to quotients of the knot. There can, however, be a finite number of surgeries for which there are exceptional additional symmetries. The included proof of this result follows the proof of Thurston's Dehn surgery theorem. The paper also includes examples of such exceptional symmetries. Since the quotients follow the behavior of knots, a census of the behavior for knots with less than 11 crossings is included.